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1

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W1 Point collocation

W1 Subdomain collocation

Exact

Subdomain collocation

Galerkin

Pointcollocation

2.5

2.0

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01.00 x/L

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W1 W2Point collocation

W1

W2

1

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Subdomain collocation

Point collocation

Galerkin

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0.3

0.2

0.1

0

Two-term approximation

One-termapproximation

t

x

y

2c

16 Q

0

(x =

0, y

= 0

, t)

22k

L2c= 1.5

= 1

Q = Q0 e t

= 0, t = 0

- * $ +

= ! & ( (

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') ( ( (

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Plane of strata parallel to xz

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E

y = 1

Exact(x = xy = 0) = 0.15

l

Nodal averages

Value atcentres oftriangles

y

x0.440.070.07

0.440.070.07

0.860.070.07

0.000.070.07

0.450.07

0.08

0.450.060.06

0.860.070.08

0.000.070.06

0.480.080.06

0.480.060.08

0.000.090.09

0.830.04

0.04

+

!

*

x = 1

x

y

! " " "#

1.0 1.0

3.0 2.83

x x

1.01.75y y

(a) Isotropic (b) Orthotropic

Exact solution for infinite plate

Finite element solution

$ % #&

52100

47100

49

48100

x

y

Restrained in y direction from movement

A

B

C

' "" &" () * $ ! !

,- ! ! + .+ "

( ( (

A*

5

(

(

(

2 '7

/ ' #0 $ .+ $

* / )

>

=

>A<

8 '''

8&

A

(

0.370.71

0.270.73

0.110.75

0.401.06

0.490.99

0.560.89

0.552.03

0.510.27

1.890.88

0.667.49

0.231.35

0.321.15

1.950.27

2.200.36

1.780.14

2.080.13

4.500.27

2.610.38

1.860.02

0.120.01

1.580.03

1.440.04 1.10

0.05

0.980.05

1.050.05

1.150.12

1.240.16

1.030.01

1.250.04

0.970.04

0.870.06

0.970.00

0.910.04

0.860.06

0.860.13

0.780.17

0.960.12

0.920.08

0.860.03

0.960.07

1.090.01

0.970.02

0.260.02

0.670.02 0.83

0.04 0.810.08 0.74

0.140.67

0.200.76

0.04

0.790.08

0.720.08

0.630.050.62

0.11

0.660.11

0.710.04

0.530.08

0.460.020.09

0.08

0.160.04

0.300.01

0.470.04 0.540.05

0.540.11

0.510.08

0.510.06

0.210.02

0.180.01

0.200.02

0.340.04

0.070.01 0.13

0.03

0.200.01

0.320.03

0.440.03

0.410.060.41

0.10

0.290.06

0.280.04

0.360.11

0.200.04

0.970.01

0.000.020.01

0.14

1 2 30

1

2

3

Tensile zone

10 to

nne/

m2

E2 = 1 tonne/m 2

x = 2(1 y2)

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

E1 = 10

E2 = 1

y

xu = 0 Region analysed

y = 1

! + , - ." /0

! "'' A

' ! ! "8#

4

! "%# A

8 8 ''%

! ! "8# ! "#

A

Buttress webthickness 9 ft

Constant web taper1 ft in 82 ft 6 in

6060

207

ft 0

in

A A121 ft 11 in 121 ft 11 in

10 ft

125

ft 0

in47 ft 0 in 12 ft 0 in

10 ft 6 in

Sectional plan AA(a)

Dam EC

Fault(no restraintassumed)

Altered grit E = EC110

Zero displacementsassumed

Grit EG =14 ECMudstone E = EC

12

Zero displacementsassumed

(b)

" 1 ( 23

6

1116

18 8 2237

1584

2929 39

31

60 66

41198

18

202

10

5

199

260283

2814

256

12

273

108

3965

47

71

29142

8 9

328 309

5

249

16

289

16

289 289

11 1

392

513

293288

19

317

18

327

7

337

8

355

7

375

11+17

17899

+136

485

70

+17

+884

+3792

952

849

84

13 54

8

77

22

6

4

9

33 15 66

342

817

205

282

4

43

11

14

48 52

34 25

188

18

253

15

240

8

238 2462

1846

27 78

14170

304

2

288

9

285

14

290

8

202

2

347

7

310

1721

315

13

-320

6

329

+4

211

+2010

+12

345

+23+16

30

13 +144

118

+59

9 265

+26

100

+10

358

+5

178

313

+4

7

332

879

14

333 332

23

8015

296

310

25

473

303

3954 7

5917

+3

47 1

+3

42

71

0

87

4

69

+3

62

2

61

+8

71 75

8 1667

+9

46+22

54

+17

51

+838

59+20

+31

28

+38

9729

7014 11

70

17+3

300

+200

200 lb/in2

300 lb/in2

Compression ()

Arrow indicatestension

Tensile zone

Tensile zone

300

+200

200 lb/in2

300 lb/in2

Compression ()

Arrow indicatestension

Below the foundation initial rock stresses should be superimposed

(a) (b)

# 1 % #/ + + + 3

! "

''3

''7

A

B ! "''3# C !

12+

( (

( (

)

* ( 2 ' (

Upstream

Tensile zone

Compression ( )

200lb/i

n2

400lb/in 2

Tension ( + )

Stresses exclude theoriginal rock stressesin foundation

Arrow indicatestension

Cracked zone(materialconsidered)as E = 0

Tensile zone buttension less thanprobable initialcompression in rock

1 ( 4)5 6 7 % #"8

Compression ( )

200 lb/in2Tension ( + )

Arrow indicatestension

+200

100

100 lb/in2

Temperature in this region 15F

Temperature dropslinearly here

No temperature charge in foundation

1 ( & % ! "9 : 9 "9%

5 31

2

49842

380

8

251257

62

13

3251

17

59

9

98

8

124

135

217

355 37

9

2 10

6

5

1

7

2

62

753

3

291

52

1

56

3

19

49

6

425

685

6

56 71

59 8

851

98

3

75

112

770

13

48 40

3

29

10

10

11

727

8

4

61

3

871

113

1

105

108

0

1

149

2

198

25

117

12

116

21

130

20

111101

1610

88

1

8926

1521

35

3222

54

4

352

2

61 1 48

6 498

52

49

389

7

11

92103

18

0

79

6 53

54

5

114

8

1612

27

12

67

121

7

28

1534 2614

429

14715

8

29323520374

105

40794955

5470

311440

4 x 106 lb

1.3 x 106 lbLoad effectof prestressing

Prestressing cables and gate reactions

60'

135

25' 25'10'Tailwater level

0 20 40 60 80 100 ft

0 10 20 30 40 ft

Water level

155'

+

+ ;

250000 lb/ft2

Indicates tension

x

y

+

* (

" G #

*

H 2 '

3' !

*

+

2 '&

/

) )

F

)

&8 "

#

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1

A 8

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( >< 8 1

9 .&

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5

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F

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H4

;

; 9 .38

& 9 ( ID 0 ,01 .%

' 90 ?(( " H

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% H29 1 + @;

.3

3 + J() IJ J "9;H# ! ! 87$' ..

7 = = + J() H ! "H ;#

! &7$%' ..7- + J() KL L0 9 9

( 7$- .33

. 0D 9 " " H

# ;;

.3

8 + J() *1 , ;

= 9F 9 # I 4 9

...

+ J() KL =

( $% .& .3'

+ J() KL 9

$ % 3. .3%

& L ) ! 4 .'&' + J() 1 H4 ? *

!

&' ! L .7% 9 ' ( # " 0 0# %$7- 4 .77

(

! " # $ % &

'(

" #

$ )

* $

* + ,

%

- " #

)

#

#

. ! & /

0

% & 1/ &23

#4

$

#&

.

z (v )

r (u )

m

ji

0 * " #

)4 5

##

- 1/ #4 #&

6

6

6

6

6

6

#7

z

rz (rz)

r (r)

z (z)

() r

5

$ $

%

$ 3

6

6

6

6

6

#2

0 ) /

"

6

6

#8

9

0 ' ( % & " #4 + , 9 3 :

6

6

#;

) =1/ #>

6 6

' (

5 1/ &;

#6

z

r

5

6 6

6

#

" )

)

! 1/ #

6

6

66 6 6

#

, 4 ? )

#4

1/ #7 1/ # 1/ #

3 /

4

4

$

#& @

! % ; + & ' ( )

5 % 6 .

$ /

$ * , '( ,

'(

$

,

! "

0 1/ 4

6 ##

6

6 #7

,$

6 #2$

# $ $

A B

#8

/ 4

#;

- ! &2

4#6

0

* $ % 6 $ ! &8

#

/ 1/ && $ "

#

0.85

2.492.06 2.09

1.93

2.21 2.26

0.010.70

0.46

0.340.16

0.76 0.560.32 0.01

1.932.082.242.33

2.35 2.16 2.01

0.770.40 0.12

0.77

0.40

0.112.36

2.172.01

2.33

2.20

2.04

0.13

0.37

0.80

Unitpressure

2.0

1.0

0

1.0

Exact

(a)

(b)

(c)

r

Stresses on section AAAxis of revolution

(a) (b) (c)

A

A

! " # $ % ! " # "% ! & "" "

% &

1/ &7 $ 1/ #7 4

6 6 #4$

% &

0

C " #&

100

400

200

0

200

400

600

0

50

100

r

r

t

T (C)

(a)

t

100 C 0 C32 51

6

23

340

(b)ExactTriangular averageQuadrilateral average

r

422

$ +, - . / . ! ! % ! 0& "

2p

Stress scale(arrow for tension) (a)(b)

Axis of revolution

Internalpressure p

q

! 01 % , " ! ! & " -)!

$ " #7 0

' &

" #2 A $ " #8 / C

% 2

Axis of symmetry (a)

2000 1000500

0

500

500

100020003000

1000

20003000

1000

1000

2000

3000

2000

3000CL

0

(Zero contourcoincides with

boundary)

! 2 $ . & 3 * ./ ./ ) -4. 5)6 -4 5 -)

(

" #;

, 9 '/(

"#

$ A # 7 % ;

D

$

$ A ) " =* 1/ &8> E $ 1/ ##

6

0 10 20 30 40 50 60

Distance scale (ft)

Finite element subdivision

Bottom of pile

100

ft

90 ft

p = 162000 lb

E pile = 600 pile = 0.25

E1 = 11 = 0.35

E2 = 402 = 0.30

Radius of pile = 1.5 ft

CL

p38

000

4294

052

200

0 5 10 15 20 25

Distance scale (ft)

010

0 20

0 30

0 40

0

Str

ess

scal

e (lb

/ft2)

E1

E2

Vertical stresses onhorizontal sections

Finite element solution for pile problem

Finite element solution forBossinesq problem

Exact solution for Boussinesq problem

% ! 3" ! 7 8& 9"- 5 / ,

$

&'

?5 % % 2

.% F)* G! + 5;7#

?5 % H? ? "

1@ 2 ;7#4 ! ) I: G @G+ ;#& @ $ % '!, ( $? ?C 0J

? #7E2 ;7;# 1B 5 ! 7;E2& ;7#7 .% F)* 4 @G+ ;22

!

"

! # $ %& '() * *+

! ,

! - &. &. ).. /.. 0! 1 ! &. 02 &.1 ).

! !

' &. &. &. /... ! &) ... , &. &. ).. $&..

# 3

3 4$. ! 5 6

, 4$

z

y

x

i

m

p

j

! !

4$

7 6 0)(1

$ & ( ) 4&6

$ & ( ) 4( $ )

' 6 0)+1

$4

4)

4

$

$

$

$

4+

8

$

$

$

$

$

$

$

$

$

4+

! , 4$

! $&

44

9

4:

4 4/

;

!

!

404/1

4$.

. .

.

.

. .

.

.

.

$4

. .

. .

. .

.

.

.

4$$

.

$

$

$

.

.

.

4$&

"

9 &$ 0 )&(1

. . 4$(

' "

! 01 0? 1

$ $ &

$ . . .$ . . .

$ . . .$ && . .

$ && .$ &&

4$)

! 3 0&$.1

!

3

! 4$+

! 6 0)()1

! . 4$4

@ A

B 6 0)(41C

2 ) ! "

z

y

x

(a)

z

x

y

(b)

!

! " - ' 0 D18 , 4&

, , 4( 0 1

9 2 $)

, 4) 5 E 0

each dividedthus

Two parts

"

+ 1 F , 4& ! 4

' , 4+ 44 5 8 =

- , 4+$& ' - , 44 2 5 (:+ % + $& , 4: $. ... 2 / "

B C

D

H

G

E

F

z

y

x

A

10 ft

Boundary conditions

u = v = w = 0 on ABCD

u = 0 on AEHDv = 0 on AEFB

All other boundaries free

symmetry

! # $%

45tons

5

10

5 10x

z = 0.5 ft

Section

z (T/ft2)

10

20

30

45tons

10

20

10

z = 1.75 ft

z = 3.75 ft

wE (ft2)

Computed value

Exact value

(a) (b)

5

# $% & '

9 ft (2.74 m)

3 ft (0.91 m)

R

22 ft (6.71 m)

15 ft (4.57 m)

40 ft (12.2 m)

1.5 ft (0.46 m)

Z

8 ft (2.43 m)

Prestressingsystem E = 5 x 10

6 lb/in2

3.52 x 106 kg/cm2

Concrete

= 0.15

rr

zz rz

400

500700

300

100

300

0

0300

0

400

100

700 500

200

200

400

600

700

800

1000

1200

1400

1000

800

600

500

100

200

300

400

00

200

300

100

0

0

0

100

200

200

2008001200

500

1000

200

0

0

200

600

900

1000

900

400700

800

900

1000

600

1400

1000

700

400

200

1400

1000

600

2000

700

500

300

100

" ( )

' - -

! $( 2 $+ "

? $) , 4/

(a)

(b)

# ! "

(. ... !

$+$: '

$%

$ *5 # 7 ? ?? 8F :..>: $&( '$ ((>++ $

! "

#$ % % &

' (

) )* + , % - "

%

%

. /0 1

% 2 ! + ! ,

!

!

" + ( 3 4#,

)

% 5#

+ , )

1 53

5

% 1 567 )

%

) /

58

9 %

: :! *

% . ; +58, +56,

% 1 5$

; +5$, %

% 1 5##

0 + /9 7, !

+55,&

%

1 5#3

% +55,

! " ##

9 .

#3

%

#

3

3

5#

0 ?@ ; +55,A ; +5$, +55,

% ( 4

9

! " #

%

9

5#6 ! ; +5#3, ; +5#, . !

5#8

7! 0

( ; +5#8, +5#3, %

1

# 5#$

5#5

" #

%

2

.

9

% )9

5#4

#6

$ #

! $# % %

. 9

% !

% ! + , 9 9 .

: 5#

y

x

y'

x '

Stratification

"

9 = . 9 = 9 ?; +5#5,A

% . 9

! &% #'

%

!

%5#>

! )9

1

1

531

/ 0 9

%536

)9

1

1

538

0 ; +5#$, 9

3

8 ( 838

% &' (

9 ! 2

: 5 0

! . 2 5

L

s

r

i

j

k

m

y

x

qn

"

6 ' C = . 2 9

8 :

% 9

% . : 58 %

2

#

#

3 1 53$

(a) (b)

2042(2085)

3041(3095)

3352(3400)

3123(3190)

4793(4850)

5317(5380)

3656(3695)

5684(5745)

6332(6405)

3818(3855)

5957(6027)

6644(6715)

1921 3132 3251

3214 4712 5397

3556 5764 6247

3914 5876 6728

" $

!" !# $ "%

600

1200

1400

1800

229 444 547 608631 623

563495 475 438

397 350 290 218 99

217

289

347

391

421

436

43587113001740

1292

855

806

725

621

480

366609 4807358329651113116011451063888707 895445

351

654

671

676

670

666 1240 1605 1743

1617

1235

1216

1170

1073

8901285 1485 1550 1540 1463 1378 1347 1260 1143 989

765

983

1132

1231

1740

14891740

1430

1740

1740

14321150

1741174317431743

1743

1743

1743

1611

1775

1580

1505

1610

17651712

200

400

8001000

1600

1743

L

L

G2/G1 = 3

G2 G1

L/2

CL

= 0 on external boundary

" %

" !&'

.

#1 % . 2 4

/ 9 ) : 5$

'

1 535

= +, 9 %

9 %

0 9

)

$# #

% . ) %

; +535, '

B =

: 55

$() *&

% 9 ) +536, +538, B

: 54

#

+)%%) #

"

)

>

%

=

H = 0H = 100

Computed values

Equipotentials of exact solution

Stratificationkx = 4ky

94.1

94.0

93.694.2

93.794.4

89.389.2

89.3

91.6

89.2

89.6

89.4

85.1

85.3

85.2

85.0

80.5

85.2

81.0

76.3

71.775.0

76.5

80.9

76.3

80.6

76.0

80.8

80.5

76.0 70.7

70.4

70.2

69.7

69.762.1

71.7

66.7

63.0

62.9

62.5

61.9

61.5

60.350.3

50.4

49.9

50.3

50.8

50.7

51.2

58.3 52.8

50.0

41.7

35.233.3

35.3

35.0

35.0

34.8

33.2

32.6

34.0

18.618.4

79.6

20.2

19.8

20.2

19.8

20.2

25.0

16.7

8.4

7.7

8.2

7.6

8.68.9

8.27.8

ku

" ) (

) * +

9

"

3 1

# 5

% +63 , ' + ( ,

DamH = 100 H = 0

95.6 90.5

77.767.5

56.548.9

36.734.726.1

19.2

11.291.1 83.5

70.6

71.1 59.5

49.441.7

42.3

35.929.2

22.5

16.9

6.79.7 8.9

7.27.7

10.015.817.2

13.1

18.6

21.5

24.322.8

20.0

30.030.0

31.4

26.7

27.829.0

24.329.4

35.7

32.7

31.2

32.7 33.3

34.2

40.833.1

33.9

36.2

33.7

36.340.0

35.1

47.640.8

55.5

54.345.3

41.6

39.3

35.634.3

34.1

34.235.0

37.1

39.2

40.0

43.9

47.741.5

45.4

50.4

56.5

58.8

53.9

50.7

60.7

70.1

79.4

71.8

79.3

81.087.3

92.2

90.791.2

83.5

84.3

83.2

90.0

89.1

80.0

77.5

73.1

71.1

76.0

69.1

61.7

63.768.770.0

66.0

60.050.0

k = 4

k = 1

k = 2

k = 1

k = 1

k = 4

k = 9

k = 3

ImperviousEquipotentials

" ( +

CL0 0 0 0 0 0

0

0 0

0

0 0

0

0 0

0

0

0

0100

100

100

100

100

100

100

100

100

100

100100100

31.0 31.229.7

41.4

30.6

13.6

22.4

30.7

14.2

13.3

34.6

10.8

8.3

36.5

8.434.3

29.2

30.263.4

65.2

64.7

64.3

65.6

64.7

62.4

54.9

62.7

63.5

63.864.464.5

Axi

s of

rot

atio

nal s

ymm

etric

al

1020

40

6080

90

6.7

23.6

13.8

37.3

" * + *+

H

Moving wall

14 21 28 35 42 49 56

5554

5352

5150

Element subdivision

1234567

11111

12

12

L = H/6x (an) L

" + ,

-

+

%

: 5#1%

# ! 9

#1

:

+ #, 1

: 5#1

)

; +534, %

#>

: 5## 6 '

!

$

.

=

9###

! % #'

.

% $56

. " +: 5#3,

: 5#

9 . $ #>$$ % ! !

#6#5

28 35 42 49 560

0.2

0.4

0.6

0.8

14 21

Exact solution(Westergaard)

Constantacceleration

Linearly varyingacceleration

p/ Ha0a0

a0

1

2

3

4

5

6

7

" , ,

%

2

0 #431

Non-conducting

Non-conducting

Typical volume element

q Specified on this face

(a)

R = 1.27 H

6748'

= constant1.25 H

H

4 H

CL

= 0

= 1922'

= 3844'

= 586'

= 586'

= 6748'

ZH

p1a1 gH

(b)

Present solution

Electrolytic tank solution

Excess pressure

Relative acceleration

Density

p1a1

" , .

. D9

% 151

. . 0 . . )9 + , 0 . /9 :

90

70

50

3010

a

ce

90

70

50

30

10

f

b

d

" '

% . 9 ; +51, # % . &

#6

% 5#

1

!

; +51,

% 153

.

5

" ! ( $ /#%

. . ; +5$, .

%% 1 56 1 . %

?; +5$,A .

/ = 9

9

' . 0 ; +5#$, +5#5,

% 58

%

%

5$

/ % 1

!

: 5#6 ' ! 4 #4

9

3 3 3

55

%

E 3 . 9

>

, -' #'

; +54, @

D

+0 .,

!

333>

CL

CL

Coil

Element subdivision

B'

A'A

B

5.260.21

0.85

5000.0

4000.0

3000.0

2000.0

1000.0

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0x (cm)

2.04.0

6.0

8.010.0

12.0

y (cm)

Tota

l fie

ld s

tren

gth

HA

-t/m

(a)

Field due tocurrent HsTotal field H(with magnetization)

300.0

200.0

100.0

0.0 4.0 cm

y

x

z

Sca

lar

pote

ntia

l A

mp-

t

(b)

" $ $% ( $% '&

.

% *&

%

" ) ) ) ) 2 % 9 9 : 9 D1 #$

2 9 )

1 5>

% 9

561

)

? ; +53,A

.

% 1 56#

L

O

CL CL

Inlet

U

2.625 in 2.375 in1.

182

in

Contours of ph21/6UL

h = 116 in116 in

" % ,

% 1 563/ ) .

56

. %

% 1 566

) ) /

9

9 E

% )

54

/

) ) %

9 ) !

%

. 9 C ) @?: 5#$+,A ) ?: 5#$+,A

!

. : ; +535,

(a)

(b)

p = 0

p = 0

p = 0

x

y

" ) +

+ & $% 0 . $%

%

1 568

.

0

9 /

56$

) . )

+0,

565% (

/ ! ; +565,

. @ ! @ / !

% ->

/ 61C64

- .

9 9 F

.

/(

# $8

3 E

/ . ! / : % /: -

$8

88

/(

4 J: ; $1

> $

#1 'D

! " !

6#4C #>## C5# ( #>$>

#3 F; B :

% % + F' 7 J% 58

# // -! /

F * F% ( %

3#5C64 #>$>#6 ( DEI B . * B 5 8C4> #>51

#8 ( D( ' : 9 . #>5#

#$ -' D* / 9 : .

D%%31 B #3 #>53#5 ; D . " ' ( 346 /

#>5

#4 C8$ #>55

#> J (! % 5>31 J (! % % .

-+$, #>41

3# *E % F ! + F B * #>$$

33 DD F :

" ! +( :, 836 #>$>3 DD F %H :

" ! +( :, 6>8 #>5136 J' / * ( %

# $ "

! #166C$ #>$>38 J: -! I' ' / . &

# , " ! +( :, # #>53

3$ I' ' / . # * #>C$4 #>56

35 (D F I

3 J' / 7 D0! * ( % ) . # $ " ! >$#C6 #>$>

"J * / . ) # * 36C83 #>51

6 7 E *' B % .

) # " " ! >54C413 #>5#8 (%I -; "! "F ' : ) ) .

" # 'H$ #>5

$ -; "! J 00 # % 65>C4 #>$>

5 ( * : ) % %+ F' 7, E # #85C43 #>58

4 J / / . ) ! 36#C8# #>$4

> F" % - - * ) " !

'H3 38C #>$561 J "! / ) # % >8C5

#>85

6# I

0 0 3

#>58

63 J - / ) )

" 1 $ $8C41 #>416 - E E D K. ) " " 8#C$1 #>58

66 JD (

J - : "! ;D8 #1>>C

#### #>5468 B I! ( ) # * " 34C>1 #>55

6$ ( * : . ) # * #6#8C#4 #>5$

65 ( * 7 " / )

" 1 $ 35C61 #>464 IJ - D I :

# * " ) #C33 #>5>

/(

!

"

# " # #

# $ % &' &(' &') " *

"

+,

&' & ' -

.

&' / & '

" "

& ('

"

" / # " / % 0

1

" /

2 + + # / 3 456 # # * /

"

+ ( # / &' # % &+3' &((' 7

3

83

/

* # / /

8+" 46

#

3 + & ' ,

3

- - 8(

-

3

3 8

" / 3

/ # 3 "

! / # 46 9 #

&83' "

" & ' 7 83 1 / & '

# & 5 '

"

7

333 3 8

7

33 3+ -

+3 ++

+(

- (+ ((

3

+

(

3

+

(

8

5 9 $% &+3(' 9 / )

* 46 # & ' " 0 7 83

:

33 33 8 7 0 9 5

33 - -

- ++ -

- - ((

3+(

3

+

(

88

" 46

! ; & ' "

"

1 / # &'

"

5 " # #

!"

# $ %

& / ' / # 7 8+ 3 8

/ & ' 1 - =

>

#

" " # ; ?

#

2

" #

#

7 9 #

3 + ( + + ( 8( 8@"

# "

A # # +

2 " # % &'

3

8

3 3 3 3 3 +3 3

+3

(3 3

(3

3 8 8 8 (8

3

8

83-

8337

3 83+ % &8@'

3 83(

3 + + ( ( 83"

3+ 8 83

3 83"

B

#+( 9 # #

C

y

x

1

2

3

4

8

7

6

5

$ %

% &83' 7 #

3 -

/ D/ 7 & #' " 7 8( " #

9

5 2 5 7 8 3 ,

83

B / 5 #

0

1

1

N1

N2

1

2

y

x

#